Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 6

Chapter 2, Problem 6

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | ≥ 7

Verified step by step guidance

Understand the inequality \(|x| \geq 7\). This means the distance of \(x\) from 0 on the number line is greater than or equal to 7.

Rewrite the inequality without the absolute value by considering the definition: \(|x| \geq 7\) is equivalent to \(x \leq -7\) or \(x \geq 7\).

Interpret the solution set as two separate intervals: one interval includes all values less than or equal to \(-7\), and the other includes all values greater than or equal to \(7\).

Look for the graph in Column II that shows two rays extending outward from \(-7\) to the left and from \(7\) to the right, including the points \(-7\) and \(7\) (usually indicated by solid dots or closed circles).

Match the inequality \(|x| \geq 7\) with the graph that represents these two intervals combined, showing the solution set correctly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequalities

Absolute value inequalities involve expressions like |x| ≥ a, which represent the distance of x from zero on the number line. The inequality |x| ≥ 7 means x is at least 7 units away from zero, so the solution includes values less than or equal to -7 and greater than or equal to 7.

Graphing Solution Sets on the Number Line

Graphing solution sets for inequalities involves shading regions on the number line that satisfy the inequality. For |x| ≥ 7, the graph shows two rays extending left from -7 and right from 7, indicating all values outside the interval (-7, 7).

Inequality Notation and Interpretation

Understanding inequality symbols (≥, ≤, >, <) is essential to interpret solution sets correctly. The symbol ≥ means 'greater than or equal to,' so values equal to or beyond the boundary points satisfy the inequality, affecting how the graph is drawn and matched.

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